no. 2
Hochschild homology and cohomology of generalized Weyl algebras
[Homologie et cohomologie de Hochschild des algèbres de Weyl généralisées]
Farinati, Marco A.1 ; Solotar, Andrea L.1 ; Suárez-Álvarez, Mariano1
1 Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellon I, 1428 Buenos Aires (Argentine)
Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 465-488.

Nous calculons l'homologie et la cohomologie de Hochschild des algèbres de Weyl généraliseés introduites par V. Bavula. Nous répondons à une question de Bavula-Jordan concernant les générateurs du groupe d'automorphismes d'une telle algèbre. Ce calcul explique aussi des résultats connus sur les invariants des algèbres de Weyl et des quotients primitifs.

We compute Hochschild homology and cohomology of a class of generalized Weyl algebras, introduced by V. V. Bavula in St. Petersbourg Math. Journal, 4 (1) (1999), 71-90. Examples of such algebras are the n-th Weyl algebras, 𝒰(𝔰𝔩 2 ), primitive quotients of 𝒰(𝔰𝔩 2 ), and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of Bavula–Jordan (Trans. A.M.S., 353 (2) (2001), 769-794) concerning the generators of the group of automorphisms of a generalized Weyl algebra. We also explain previous results on the invariants of Weyl algebras and of primitive quotients

DOI : 10.5802/aif.1950
Classification : 16E40, 17B37, 16S32
Keywords: Hochschild cohomology, generalized Weyl algebras, automorphism group
Mot clés : cohomologie de Hochschild, algèbres de Weyl généralisées, groupe d'automorphismes

Farinati, Marco A. 1 ; Solotar, Andrea L. 1 ; Suárez-Álvarez, Mariano 1

1 Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellon I, 1428 Buenos Aires (Argentine) 
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Farinati, Marco A.; Solotar, Andrea L.; Suárez-Álvarez, Mariano. Hochschild homology and cohomology of generalized Weyl algebras. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 465-488. doi : 10.5802/aif.1950. https://aif.centre-mersenne.org/articles/10.5802/aif.1950/

[1] J. Alev; M. Farinati; T. Lambre; A. Solotar Homologie des invariants d'une algèbre de Weyl sous l'action d'un groupe fini, J. of Alg., Volume Vol. 232 (2000) no. 2, pp. 564-577 | DOI | MR | Zbl

[2] J. Alev; T. Lambre Comparaison de l'homologie de Hochschild et de l'homologie de Poisson pour une déformation des surfaces de Klein, Algebra and Operator Theory, Proceedings of the Colloquium, Tashkent (1998), pp. 25-38 | Zbl

[3] J. Alev; T. Lambre Homologie des invariants d'une algèbre de Weyl, K-Theory, Volume 18 (1999), pp. 401-411 | DOI | MR | Zbl

[4] V. Bavula Generalized Weyl algebras and their representations, St. Petersbourg Math. J, Volume 4 (1990) no. 1, pp. 71-90 | MR | Zbl

[5] V. Bavula; D. Jordan Isomorphism problems and groups automorphisms for generalized Weyl algebras, Trans. Am. Math. Soc, Volume 353 (2001) no. 2, pp. 769-794 | DOI | MR | Zbl

[6] V. Bavula; T. Lenagan Krull dimension of generalized Weyl algebras with noncommutative coefficients, J. of Alg, Volume 235 (2001) no. 1, pp. 315-358 | DOI | MR | Zbl

[7] D. Burghelea; M. Vigué-Poirrier Cyclic homology of commutative algebras, I, Springer Lecture Notes in Math, Volume 1318 (1988), pp. 51-72 | DOI | MR | Zbl

[8] H. Cartan; S. Eilenberg Homological algebra, Princeton Univ. Press, Princeton, 1956 | MR | Zbl

[9] J. Dixmier Quotients simples de l'algèbre enveloppante de 𝔰𝔩 2 , J. of Alg., Volume 24 (1973), pp. 551-574 | MR | Zbl

[10] P. Etingof; V. Ginzburg Symplectic reflection algebras, Calogero-Moser spaces, and deformed Harish-Chandra homomorphism (e-print, arXiv:math.AG/0011114 v5) | MR | Zbl

[11] O. Fleury Automorphismes d'algèbres enveloppantes classiques et quantifiées : sous-groupes finis et invariants (1997) (Thèse Université de Reims, Champagne-Ardenne)

[12] O. Fleury Sur les invariants de B λ sous l'action de sous-groupes finis d'automorphismes: conjecture de Gelfand - Kirillov et homologie de Hochschild, Comm. in Alg, Volume 29 (2001) no. 8, pp. 3535-3557 | DOI | MR | Zbl

[13] T.J. Hodges Noncommutative deformations of type-A Kleinian singularities, J. Algebra, Volume 161 (1993) no. 2, pp. 271-290 | DOI | MR | Zbl

[14] C. Kassel L'homologie cyclique des algèbres enveloppantes, Invent. Math, Volume 91 (1988) no. 2, pp. 221-251 | DOI | EuDML | MR | Zbl

[15] C. Kassel; M. Vigué-Poirrier Homologie des quotients primitifs de l'algèbre enveloppante de 𝔰𝔩 2 , Math. Ann, Volume 294 (1992) no. 3, pp. 483-502 | EuDML | MR | Zbl

[16] S. Smith A class of algebras similar to the enveloping algebra of 𝔰𝔩 2 , Trans. Am. Math. Soc, Volume 322 (1990) no. 1, pp. 285-314 | MR | Zbl

[17] T. A. Springer Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, Berlin-Heidelberg-New York, 1977 | MR | Zbl

[18] R. Sridharan Filtered algebras and representations of Lie algebras, Trans. Am. Math. Soc, Volume 100 (1961), pp. 530-550 | DOI | MR | Zbl

[19] D. Stefan Hochschild cohomology on Hopf-Galois extensions, J. Pure Appl. Alg, Volume 103 (1995) no. 2, pp. 221-233 | DOI | MR | Zbl

[20] M. Suárez-Álvarez Multiplicative structure of Hochschild cohomology of the ring of invariants of a Weyl algebra under finite groups (To appear in J. of Alg) | MR | Zbl

[21] M. Suárez-Álvarez Hochschild cohomology of primitive quotients of U ( 𝔰𝔩 2 ) and their rings of invariants (Preprint, http://www.math.jussieu.fr/\mariano)

[22] M. Van den Bergh A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Am. Math. Soc, Volume 126 (1998) no. 5, pp. 1345-1348 | DOI | MR | Zbl

[23] M. Van den Bergh A relation between Hochschild homology and cohomology for Gorenstein rings (Erratum) (to appear) | MR | Zbl

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